12.21.8 problem section 10.4, problem 8

Internal problem ID [2246]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.4, constant coefficient homogeneous system. Page 540
Problem number : section 10.4, problem 8
Date solved : Monday, January 27, 2025 at 05:44:03 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=y_{1} \left (t \right )-y_{2} \left (t \right )-2 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=y_{1} \left (t \right )-2 y_{2} \left (t \right )-3 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-4 y_{1} \left (t \right )+y_{2} \left (t \right )-y_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 71

dsolve([diff(y__1(t),t)=1*y__1(t)-1*y__2(t)-2*y__3(t),diff(y__2(t),t)=1*y__1(t)-2*y__2(t)-3*y__3(t),diff(y__3(t),t)=-4*y__1(t)+1*y__2(t)-1*y__3(t)],singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{2 t} c_1 +c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{-3 t} \\ y_{2} \left (t \right ) &= {\mathrm e}^{2 t} c_1 +4 c_2 \,{\mathrm e}^{-t}+2 c_3 \,{\mathrm e}^{-3 t} \\ y_{3} \left (t \right ) &= -{\mathrm e}^{2 t} c_1 -c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{-3 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 102

DSolve[{D[ y1[t],t]==1*y1[t]-1*y2[t]-2*y3[t],D[ y2[t],t]==1*y1[t]-2*y2[t]-3*y3[t],D[ y1[t],t]==-4*y1[t]+1*y2[t]-1*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to \frac {1}{576} e^{-3 t} \left (c_1 \left (63-128 e^t\right )+c_2 \left (64 e^t-27\right )\right ) \\ \text {y2}(t)\to \frac {1}{864} e^{-3 t} \left (c_2 \left (224 e^t-81\right )-7 c_1 \left (64 e^t-27\right )\right ) \\ \text {y3}(t)\to \frac {e^{-3 t} \left (c_1 \left (189-128 e^t\right )+c_2 \left (64 e^t-81\right )\right )}{1728} \\ \end{align*}